TG01PD

Bi-domain spectral splitting of a subpencil of a descriptor system

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

  To compute orthogonal transformation matrices Q and Z which
  reduce the regular pole pencil A-lambda*E of the descriptor system
  (A-lambda*E,B,C) to the generalized real Schur form with ordered
  generalized eigenvalues. The pair (A,E) is reduced to the form

             ( *  *  *  * )             ( *  *  *  * )
             (            )             (            )
             ( 0  A1 *  * )             ( 0  E1 *  * )
    Q'*A*Z = (            ) ,  Q'*E*Z = (            ) ,
             ( 0  0  A2 * )             ( 0  0  E2 * )
             (            )             (            )
             ( 0  0  0  * )             ( 0  0  0  * )

  where the subpencil A1-lambda*E1 contains the eigenvalues which
  belong to a suitably defined domain of interest and the subpencil
  A2-lambda*E2 contains the eigenvalues which are outside of the
  domain of interest.
  If JOBAE = 'S', the pair (A,E) is assumed to be already in a
  generalized real Schur form and the reduction is performed only
  on the subpencil A12 - lambda*E12 defined by rows and columns
  NLOW to NSUP of A - lambda*E.

Specification
      SUBROUTINE TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P,
     $                   NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB,
     $                   C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR, ALPHAI,
     $                   BETA, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER        COMPQ, COMPZ, DICO, JOBAE, STDOM
      INTEGER          INFO, LDA, LDB, LDC, LDE, LDQ, LDWORK, LDZ, M, N,
     $                 NDIM, NLOW, NSUP, P
      DOUBLE PRECISION ALPHA
C     .. Array Arguments ..
      DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
     $                 BETA(*),  C(LDC,*),  DWORK(*),  E(LDE,*),
     $                 Q(LDQ,*), Z(LDZ,*)

Arguments

Mode Parameters

  DICO    CHARACTER*1
          Specifies the type of the descriptor system as follows:
          = 'C':  continuous-time system;
          = 'D':  discrete-time system.

  STDOM   CHARACTER*1
          Specifies whether the domain of interest is of stability
          type (left part of complex plane or inside of a circle)
          or of instability type (right part of complex plane or
          outside of a circle) as follows:
          = 'S':  stability type domain;
          = 'U':  instability type domain.

  JOBAE   CHARACTER*1
          Specifies the shape of the matrix pair (A,E) on entry
          as follows:
          = 'S':  (A,E) is in a generalized real Schur form;
          = 'G':  A and E are general square dense matrices.

  COMPQ   CHARACTER*1
          = 'I':  Q is initialized to the unit matrix, and the
                  orthogonal matrix Q is returned;
          = 'U':  Q must contain an orthogonal matrix Q1 on entry,
                  and the product Q1*Q is returned.
                  This option can not be used when JOBAE = 'G'.

  COMPZ   CHARACTER*1
          = 'I':  Z is initialized to the unit matrix, and the
                  orthogonal matrix Z is returned;
          = 'U':  Z must contain an orthogonal matrix Z1 on entry,
                  and the product Z1*Z is returned.
                  This option can not be used when JOBAE = 'G'.

Input/Output Parameters
  N       (input) INTEGER
          The number of rows of the matrix B, the number of columns
          of the matrix C, and the order of the square matrices A
          and E.  N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix B.  M >= 0.

  P       (input) INTEGER
          The number of rows of the matrix C.  P >= 0.

  NLOW,   (input) INTEGER
  NSUP    (input) INTEGER
          NLOW and NSUP specify the boundary indices for the rows
          and columns of the principal subpencil of A - lambda*E
          whose diagonal blocks are to be reordered.
          0 <= NLOW <= NSUP <= N,       if JOBAE = 'S'.
          NLOW = MIN( 1, N ), NSUP = N, if JOBAE = 'G'.

  ALPHA   (input) DOUBLE PRECISION
          The boundary of the domain of interest for the generalized
          eigenvalues of the pair (A,E). For a continuous-time
          system (DICO = 'C'), ALPHA is the boundary value for the
          real parts of the generalized eigenvalues, while for a
          discrete-time system (DICO = 'D'), ALPHA >= 0 represents
          the boundary value for the moduli of the generalized
          eigenvalues.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the state dynamics matrix A.
          If JOBAE = 'S' then A must be a matrix in real Schur form.
          On exit, the leading N-by-N part of this array contains
          the matrix Q'*A*Z in real Schur form, with the elements
          below the first subdiagonal set to zero.
          The leading NDIM-by-NDIM part of the principal subpencil
          A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP)
          and E12 := E(NLOW:NSUP,NLOW:NSUP), has generalized
          eigenvalues in the domain of interest, and the trailing
          part of this subpencil has generalized eigenvalues outside
          the domain of interest.
          The domain of interest for eig(A12,E12), the generalized
          eigenvalues of the pair (A12,E12), is defined by the
          parameters ALPHA, DICO and STDOM as follows:
            For DICO = 'C':
               Real(eig(A12,E12)) < ALPHA if STDOM = 'S';
               Real(eig(A12,E12)) > ALPHA if STDOM = 'U'.
            For DICO = 'D':
               Abs(eig(A12,E12)) < ALPHA if STDOM = 'S';
               Abs(eig(A12,E12)) > ALPHA if STDOM = 'U'.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= MAX(1,N).

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,N)
          On entry, the leading N-by-N part of this array must
          contain the descriptor matrix E.
          If JOBAE = 'S', then E must be an upper triangular matrix.
          On exit, the leading N-by-N part of this array contains an
          upper triangular matrix Q'*E*Z, with the elements below
          the diagonal set to zero.
          The leading NDIM-by-NDIM part of the principal subpencil
          A12 - lambda*E12 (see description of A) has generalized
          eigenvalues in the domain of interest, and the trailing
          part of this subpencil has generalized eigenvalues outside
          the domain of interest.

  LDE     INTEGER
          The leading dimension of the array E.  LDE >= MAX(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading N-by-M part of this array must
          contain the input matrix B.
          On exit, the leading N-by-M part of this array contains
          the transformed input matrix Q'*B.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= MAX(1,N).

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the leading P-by-N part of this array must
          contain the output matrix C.
          On exit, the leading P-by-N part of this array contains
          the transformed output matrix C*Z.

  LDC     INTEGER
          The leading dimension of the array C.  LDC >= MAX(1,P).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          If COMPQ = 'I':  on entry, Q need not be set;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix Q,
                           where Q' is the product of orthogonal
                           transformations which are applied to A,
                           E, and B on the left.
          If COMPQ = 'U':  on entry, the leading N-by-N part of this
                           array must contain an orthogonal matrix
                           Q1;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix
                           Q1*Q.

  LDQ     INTEGER
          The leading dimension of the array Q. LDQ >= MAX(1,N).

  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
          If COMPZ = 'I':  on entry, Z need not be set;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix Z,
                           which is the product of orthogonal
                           transformations applied to A, E, and C
                           on the right.
          If COMPZ = 'U':  on entry, the leading N-by-N part of this
                           array must contain an orthogonal matrix
                           Z1;
                           on exit, the leading N-by-N part of this
                           array contains the orthogonal matrix
                           Z1*Z.

  LDZ     INTEGER
          The leading dimension of the array Z. LDZ >= MAX(1,N).

  NDIM    (output) INTEGER
          The number of generalized eigenvalues of the principal
          subpencil A12 - lambda*E12 (see description of A) lying
          inside the domain of interest for eigenvalues.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
  BETA    (output) DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N,
          are the generalized eigenvalues.
          ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1,...,N, are the
          diagonals of the complex Schur form (S,T) that would
          result if the 2-by-2 diagonal blocks of the real Schur
          form of (A,B) were further reduced to triangular form
          using 2-by-2 complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real;
          if positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

Workspace
  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= 8*N+16, if JOBAE = 'G';
          LDWORK >= 4*N+16, if JOBAE = 'S'.
          For optimum performance LDWORK should be larger.

          If LDWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the DWORK
          array, returns this value as the first entry of the DWORK
          array, and no error message related to LDWORK is issued by
          XERBLA.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the QZ algorithm failed to compute all generalized
                eigenvalues of the pair (A,E);
          = 2:  a failure occured during the ordering of the
                generalized real Schur form of the pair (A,E).

Method
  If JOBAE = 'G', the pair (A,E) is reduced to an ordered
  generalized real Schur form using an orthogonal equivalence
  transformation A <-- Q'*A*Z and E <-- Q'*E*Z. This transformation
  is determined so that the leading diagonal blocks of the resulting
  pair (A,E) have generalized eigenvalues in a suitably defined
  domain of interest. Then, the transformations are applied to the
  matrices B and C: B <-- Q'*B and C <-- C*Z.
  If JOBAE = 'S', then the diagonal blocks of the subpencil
  A12 - lambda*E12, defined by A12 := A(NLOW:NSUP,NLOW:NSUP)
  and E12 := E(NLOW:NSUP,NLOW:NSUP), are reordered using orthogonal
  equivalence transformations, such that the leading blocks have
  generalized eigenvalues in a suitably defined domain of interest.

Numerical Aspects
                                  3
  The algorithm requires about 25N  floating point operations.

Further Comments
  None
Example

Program Text

*     TG01PD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      DOUBLE PRECISION ZERO
      PARAMETER        ( ZERO = 0.0D0 )
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, MMAX, PMAX
      PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDB, LDC, LDE, LDQ, LDZ
      PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
     $                   LDE = NMAX, LDQ = NMAX, LDZ = NMAX )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = 8*NMAX+16 )
*     .. Local Scalars ..
      CHARACTER*1      COMPQ, COMPZ, DICO, JOBAE, STDOM
      INTEGER          I, INFO, J, M, N, NDIM, NLOW, NSUP, P
      DOUBLE PRECISION ALPHA, TOL
*     .. Local Arrays ..
      DOUBLE PRECISION A(LDA,NMAX),  ALPHAI(NMAX), ALPHAR(NMAX),
     $                 B(LDB,MMAX),    BETA(NMAX), C(LDC,NMAX),
     $                 DWORK(LDWORK), E(LDE,NMAX), Q(LDQ,NMAX),
     $                 Z(LDZ,NMAX)
*     .. External Functions ..
      LOGICAL          LSAME
      EXTERNAL         LSAME
*     .. External Subroutines ..
      EXTERNAL         TG01PD
*     .. Intrinsic Functions ..
      INTRINSIC        DCMPLX
*     .. Executable Statements ..
*
      WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
      READ ( NIN, FMT = '()' )
      READ ( NIN, FMT = * ) N, M, P, DICO, STDOM, JOBAE, COMPQ, COMPZ,
     $                      NLOW, NSUP, ALPHA, TOL
      IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99988 ) N
      ELSE
         READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
         READ ( NIN, FMT = * ) ( ( E(I,J), J = 1,N ), I = 1,N )
         IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
            WRITE ( NOUT, FMT = 99987 ) M
         ELSE
            READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
            IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
               WRITE ( NOUT, FMT = 99986 ) P
            ELSE
               READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
               IF ( LSAME( COMPQ, 'U' ) )
     $            READ ( NIN, FMT = * ) ( ( Q(I,J), J = 1,N ), I = 1,N )
               IF ( LSAME( COMPZ, 'U' ) )
     $            READ ( NIN, FMT = * ) ( ( Z(I,J), J = 1,N ), I = 1,N )
*              Find the reduced descriptor system
*              (A-lambda E,B,C).
               CALL TG01PD( DICO, STDOM, JOBAE, COMPQ, COMPZ, N, M, P,
     $                      NLOW, NSUP, ALPHA, A, LDA, E, LDE, B, LDB,
     $                      C, LDC, Q, LDQ, Z, LDZ, NDIM, ALPHAR,
     $                      ALPHAI, BETA, DWORK, LDWORK, INFO )
*
               IF ( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99994 ) NDIM
                  WRITE ( NOUT, FMT = 99997 )
                  DO 10 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
   10             CONTINUE
                  WRITE ( NOUT, FMT = 99996 )
                  DO 20 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( E(I,J), J = 1,N )
   20             CONTINUE
                  WRITE ( NOUT, FMT = 99993 )
                  DO 30 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
   30             CONTINUE
                  WRITE ( NOUT, FMT = 99992 )
                  DO 40 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
   40             CONTINUE
                  WRITE ( NOUT, FMT = 99991 )
                  DO 50 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( Q(I,J), J = 1,N )
   50             CONTINUE
                  WRITE ( NOUT, FMT = 99990 )
                  DO 60 I = 1, N
                     WRITE ( NOUT, FMT = 99995 ) ( Z(I,J), J = 1,N )
   60             CONTINUE
                  WRITE ( NOUT, FMT = 99985 )
                  DO 70 I = 1, N
                     IF ( BETA(I).EQ.ZERO .OR. ALPHAI(I).EQ.ZERO ) THEN
                        WRITE ( NOUT, FMT = 99984 )
     $                     ALPHAR(I)/BETA(I)
                     ELSE
                        WRITE ( NOUT, FMT = 99984 )
     $                     DCMPLX( ALPHAR(I), ALPHAI(I) )/BETA(I)
                     END IF
   70             CONTINUE
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' TG01PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TG01PD = ',I2)
99997 FORMAT (/' The transformed state dynamics matrix Q''*A*Z is ')
99996 FORMAT (/' The transformed descriptor matrix Q''*E*Z is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (' Number of eigenvalues in the domain =', I5)
99993 FORMAT (/' The transformed input/state matrix Q''*B is ')
99992 FORMAT (/' The transformed state/output matrix C*Z is ')
99991 FORMAT (/' The left transformation matrix Q is ')
99990 FORMAT (/' The right transformation matrix Z is ')
99988 FORMAT (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' M is out of range.',/' M = ',I5)
99986 FORMAT (/' P is out of range.',/' P = ',I5)
99985 FORMAT (/' The finite generalized eigenvalues are '/
     $         ' real  part     imag  part ')
99984 FORMAT (1X,F9.4,SP,F9.4,S,'i ')
      END
Program Data
TG01PD EXAMPLE PROGRAM DATA
  4     2     2     C     S     G     I     I     1     4     -1.E-7     0.0    
    -1     0     0     3
     0     0     1     2
     1     1     0     4
     0     0     0     0
     1     2     0     0
     0     1     0     1
     3     9     6     3
     0     0     2     0
     1     0
     0     0
     0     1
     1     1
    -1     0     1     0
     0     1    -1     1
Program Results
 TG01PD EXAMPLE PROGRAM RESULTS

 Number of eigenvalues in the domain =    1

 The transformed state dynamics matrix Q'*A*Z is 
  -1.6311   2.1641  -3.6829  -0.3369
   0.0000   0.4550  -1.9033   0.6425
   0.0000   0.0000   2.6950   0.6882
   0.0000   0.0000   0.0000   0.0000

 The transformed descriptor matrix Q'*E*Z is 
   0.4484   9.6340  -1.2601  -5.6475
   0.0000   3.3099   0.6641  -1.4869
   0.0000   0.0000   0.0000  -1.3765
   0.0000   0.0000   0.0000   2.0000

 The transformed input/state matrix Q'*B is 
   0.0232  -0.9413
  -0.7251  -0.2478
   0.6882  -0.2294
   1.0000   1.0000

 The transformed state/output matrix C*Z is 
  -0.8621   0.3754   0.3405   1.0000
  -0.1511  -1.1192   0.8513  -1.0000

 The left transformation matrix Q is 
   0.0232  -0.7251   0.6882   0.0000
  -0.3369   0.6425   0.6882   0.0000
  -0.9413  -0.2478  -0.2294   0.0000
   0.0000   0.0000   0.0000   1.0000

 The right transformation matrix Z is 
   0.8621  -0.3754  -0.3405   0.0000
  -0.4258  -0.9008  -0.0851   0.0000
   0.0000   0.0000   0.0000   1.0000
   0.2748  -0.2184   0.9364   0.0000

 The finite generalized eigenvalues are 
 real  part     imag  part 
   -3.6375
    0.1375
  Infinity
    0.0000

Return to index